New formulas for the Jones polynomial of a rational link

TL;DR

This paper introduces new formulas for computing the Jones and Kauffman bracket polynomials of rational links, extending previous results to non-alternating diagrams and providing an automaton for crossing sign calculation.

Contribution

It generalizes existing formulas for rational links' polynomials and introduces a finite automaton for crossing sign computation in standard diagrams.

Findings

Derived new formulas for Jones and Kauffman bracket polynomials of rational links.

Extended formulas to non-alternating diagrams of rational links.

Developed a finite automaton to compute crossing signs and writhe.

Abstract

We derive new formulas for the Jones polynomial and the Kauffman bracket polynomial of a rational link represented by a standard diagram that is not necessarily alternating. These formulas generalize the results of Qazaqzeh, Yasein, and Abu-Qamar for the Tutte polynomial of the Tait graph of an alternating diagram of a rational link, as well as the matrix formulas of Lawrence and Rosenstein for the Jones polynomial of a rational link. Our approach uses the colored version of Brylawski's tensor product formula for Tutte polynomials of colored graphs, due to Diao, Hetyei, and Hinson. Furthermore, generalizing the formulas of Qazaqzeh, Yasein, and Abu-Qamar, we present a finite automaton that computes the crossing signs, thereby enabling the calculation of the writhe of a standard diagram of a rational link.